Theorem of Cauchy on Arrangements. 381 



ought to be the coefficient of A, in (A,+ l)(X-f2)(X + 3), i. e. 

 11, which is right; for the number of substitutions of the 

 form {a, b) (c, d) will be 3, and of the form («, b, c) (if) 8. In 

 conclusion, I may notice that by an obvious deduction from 

 this last theorem, we are led to the well-known one in the theory 

 of numbers, that every coefficient in the development of 



^{p + \){p + %)... {p + n-l), 



except the first and last, and the sum of these two, is divisible 

 by n when n is a prime number ; and indeed we can actually ex- 

 press by aid of it the quotient of every intermediate coefficient 

 divided by n as the sum of separate integer terms free from the 

 sign of addition. 



K, Woolwich Common, 

 October 10, 1861. 



Postscript. By an extension of the method of generating func- 

 tions contained in the text above, it may easily be seen that the 

 number* of substitutions of n letters represented by the products 

 of r cyclical substitutions, where the number of letters of each 

 cycle leaves a given residue e in respect to a given modulus /x, may 

 be made to depend on the solution of the equation in differences 



u n -u n+ „z=-^— Un _ e . 



The case where e=l is deserving of particular notice. 



It may be shown by means of the above equation in dif- 

 ferences, that the number of substitutions of n letters formed 

 by r cycles each of the form /u,K + l (/u, being constant), say 



<f)(n, r } fi, 1), where is necessarily an integer, may be 



found by taking in every possible way distinct groups of /j, 



consecutive terms of the series 1, 2, 3, . . . (n — 1) ; the sum of 

 the products of every such combination of groups is the value 

 required. For example, if 



n = 8, r=3, fi = 2, 



For this number, divided by n(w), is the coefficient of x n in 



d4f x e ~ l 

 and therefore of ir rt p r in e?<f * , say ^(x, p), and therefore (since -r- = ^ 



and ^ may be put under the form 2 — x n ) of p r in — , where u n is defined 



as in the text. 



